Frame rate matters, more or less. In a tight fight it most likely does.
30 FPS has been considered the minimum for enjoyable game play since day 1 and 60 was golden but a couple of things have happened since then. One is that the refresh rate of gaming monitors has increased. Another one is that the resolution of them has also increased. Those two combined or separate require much more calculating power than before. The third one is that AH1 and 2 relied heavily on the CPU (processor) whereas AH3 is like most other modern games and requires a decent video card. You don't need the most expensive one on the market for AH but you definitely need a good one. Posting the first third of your DxDiag output as an attachment here would help telling if that's the culprit.
The fourth thing to take into consideration is that because of the first two now having existed for quite some time, active players have adapted to them and according to tests 60 FPS may not be the upper limit for the human eye to see.